Communications ΪΠ
Commun. Math. Phys. 134, 587-617 (1990)
Mathematical
Physics
© Springer-Verlag 1990
Similarity Between the Mandelbrot Set
and Julia Sets
TAN Lei*
Institut fur Dynamische Systeme, Universitat Bremen, W-2800 Bremen 33,
Federal Republic of Germany
Received July 10, 1989
Abstract. The Mandelbrot set M is "self-similar" about any Misiurewicz point c in
the sense that if we examine a neighborhood of c in M with a very powerful
microscope, and then increase the magnification by a carefully chosen factor, the
picture will be unchanged except for a rotation. The corresponding Julia set Jc is
also "self-similar" in the same sense, with the same magnification factor. Moreover,
the two sets M and Jc are "similar" in the sense that if we use a very powerful
microscope to look at M and Jc, both focused at c, the structures we see look like
very much the same.
1. Introduction
2
For a quadratic polynomial fc : z\-*z + c, the filled-in Julia set Kc oϊfc is the set of
non-escaping points under iteration:
Kc = {z e <C I (fcn(z))ne^
n
ih
where fc denotes the n iteration f c ° f c ° . . . o f c o f f .
is bounded} ,
The Julia set of fc is Jc = dKc.
The Mandelbrot set is
One can generate easily the pictures of Julia sets and the Mandelbrot set by
computers. Figure 1 is a picture of M , Fig. 3a-3d are pictures of Jc for various
values of c. Globally, Jc and M have completely different shapes. However, their
local structures are sometimes very similar. Figure 2 consists of three successive
enlargements of M in a neighborhood of L A remarkable resemblance with the
Julia set for c = i (Fig. 3a) appears. In fact, this kind of similarity happens for every
value of c which is a Misiurewicz point, that is, for which the point 0 under fc is not
*- Present address: Ecole Normale Superieure de Lyon, 46 Allee dΊtalie, F-69364 Lyon, France
TAN Lei
588
Fig. 1
ί
*W
Fig. 2. Three successive magnifications of M as indicated by the inserted frames. Center: ί. Width
of the first picture: 0.5656854^0.1 x4]/2. Magnification factor: 4j/2
Fig.3a-d. The Julia set Jc. a c = i. Center: 0, width: 3.0. b c = 0.11031 -0.67037Ϊ. Center: 0,
width: 3.0. c= -1.25. Center: 0, width: 3.6. d c= -0.481762-0.531657i. Center: 0, width: 3.0
periodic but falls eventually into a periodic orbit. The main topic of this paper is to
analyse this similarity.
The point c = i is one of the simplest examples of Misiurewicz points: for
f t : zM>z2 + i, the orbit of 0 is: Oι-»ih-»i-1 H-> - ih-π-1. We have fi4(0)=fi2(Q). The
point c = — 2 is another one: for /_ 2 : zt->z2 — 2, the orbit of 0 is: OH-> — 2ι—>2n->2. In
Similarity Between the Mandelbrot Set and Julia Sets
589
Sect. 5 we will give further examples. Using the Montel theorem, one can show that
the set of Misiurewicz points forms a countable dense subset of the boundary of M
[M].
In this paper, we will show that for every Misiurewicz point c, a powerful
magnification of Jc focused at c and the same magnification of M focused also
at c will give very similar structures (up to a multiplication by a complex number),
and the more powerful the magnification is, the more similar the structures are. We
say that Jc and M are asymptotically similar about c. Moreover, both Jc and M are
asymptotically self-similar about c in the sense that if we increase successively the
magnifications to Jc (respectively M) focused at c by a carefully chosen factor, what
we see are more and more the same.
Section 2 consists of definitions of asymptotic similarity and asymptotic selfsimilarity, and introduces the Hausdorff-Chabauty distance in order to compare
two closed sets asymptotically. Section 3 discusses the asymptotic self-similarity of
any Julia Jc about any repulsive periodic point, and about any point which is
eventually repulsive periodic. This can be applied immediately to the case where c
is a Misiurewicz point, since the point c is eventually repulsive periodic under fc.
Section 4 contains a key proposition which makes connections of the selfsimilarity between parameter space and dynamical spaces. This proposition is
stated in a high-dimension form, in order to be applied eventually to other topics.
Section 5 shows finally the asymptotic similarity between M and Jc about c for c a
Misiurewicz point, by applying the result of Sect. 4. Theorem 5.5 is a review of the
principal steps of the whole proof. At the end of Sect. 5 we calculate some
specific examples. Section 6 consists of some related results, and a discussion
of the similarity problems about other points of M.
The main part of this work was done in 1984, under the direction of Professor
A. Douady. A preliminary version of this article ([T2]) was inserted in [DH2].
The author would like to thank A. Douady for his introduction and his
powerful support of this work; without him this work would not have existed. I
also thank J. H. Hubbard, M. Shishikura, Y. Fisher, B. Bielefeld, and B. Branner
for their helpful discussions: some of the calculations are made directly by M.
Shishikura. I am indebted to H.-O. Peitgen and his group in Bremen for hospitality
and great interest in the work.
2. Hausdorff Distance and Asymptotic Similarity
In this section we discuss the definition of asymptotic similarities of closed subsets
of (C and some immediate results.
Definition 1 (Hausdorff distance). Denote by 2F the set of non-empty compact
subsets of (C. For A,Be3F, we define the semi-distance from A to B to be
δ(A,B)= supφc,β)
xzA
and the Hausdorff distance of A, B to be
This distance makes 2F a complete metric space. By the definition, δ(A,B)<ε
signifies that A is contained in the s-neighborhood of B.
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TAN Lei
The next definition is for comparing two closed subsets of C within a bounded
region.
Definition 2. For r a positive real number, we denote by Dr the disc centered at 0 of
radius r. For any closed set Ac<C, we define a compact set
Ar = (AnB,)vdDr.
(1)
(It is for a technical reason that we add dDr to AnDr.) Or, in order to analyse the
local behavior of A near a point a eA, we use the translation τ_ α :zH->z — α to
translate a to the origin and define
For A, B two closed sets in C, we have Ar, Br e 3F . We can then define the
Hausdorff-Chabauty distance of A, B in the window of Dr to be (Fig. 4)
Definitions. Assume ρ = \ρ\eίθ, |ρ|>l, 0^
1. A closed subset B of C is ρ-self-similar about 0 if there is r>0 such that
Br.
(2)
B is also considered to be self-similar about 0 with the scale ρ. This says that under
the magnification with the factor |ρ| together with the rotation of angle θ the set B
remains unchanged within Dr.
2. A closed subset A of (C is asymptotically ρ-self-similar about a point x e A if
there is r>0 and a closed set B such that
(ρnτ_xA)r-+Br
while n-»oo
(3)
for the Hausdorff distance. The set B is automatically ρ-self-similar about 0, it is
called the limit model of A at x.
3. Two closed sets A and B are asymptotically similar about 0 if there is r > 0 such
that
lim
ίe€,ί-»oo
Fig. 4. A and Ar
dr(tA,tB) = Q.
(4)
Similarity Between the Mandelbrot Set and Julia Sets
591
Remark. The property of similarity does not depend really on the value of r. In fact
if (2), (3), or (4) hold for some value r > 0, then they hold for any 0 < τ' ^ τ, and (4)
holds for every r > 0, since ί-» oo. Moreover, if B satisfies (2) for some value r > 0, we
can set
(5)
neN
Then B'nDf = BnDr and (ρF)5 = B's for any 5 > 0. If A, B satisfy (3) for some value
r>0, then by replacing B by B', the equality (3) will hold for every value s>0
instead of r (since \ρ\n-+ao while n-»oo).
Examples
1. The disc Dr itself is trivially self-similar. A line, a segment are self-similar. A
curve with a tangent is asymptotically similar to its tangent.
2. The spiral S(λ) = {eλx|xeR} is eΛ-self-similar about 0 if \eλ\>\ (Fig. 5).
3. The Cantor set C. C is self-similar about any rational point, and is not selfsimilar about any irrational point xeC. Let x = 0.tit2t3... be the expansion of
x 6 C in base three. Then each tt is either 0 or 2. x is rational if and only if this
expansion is eventually periodic, i.e. there are / and p such that tn = tn+p for each
n ^ /. The minimal p is the eventual period of x. Hence C is 3p-self-similar at x e C if
and only if x is rational and the eventual period of x divides p.
Lemma 2.1. Assume A, B closed. Fix r>0. We have
1. For every aeAnDr) d(a,Br)<Zd(a,B)^d(a,BnDr).
2. δ(Ar,Br)^ sup d(a,B)£d(AnDnBnDr).
aeAr\Dr
Proof. Left to the reader.
Π
Proposition 2.2.
1. For A, B closed subsets of (C, if there are r>0, ρ, |ρ|>l such that
n
n
lim dr(ρ A,ρ B) = Q
(6)
lim
(7)
then
f-»oo,fe<C
dr(tA,tB) = Q,
i.e. A and B are asymptotically similar about 0.
1
0.8
06
0.4
02
0
-0.2
-0.4
-0.6
-0.8
-0.6
-0.4 -0.2
0
0.2
Fig. 5. a g(1°β1 2+ί"/4)*; b
0.4
0.6
0.8
592
TAN Lei
2. Any asymptotically self-similar set is asymptotically similar to its limit model.
Proof.
1 . For each β > 0, there is N > 0 such that if n > N then δ((ρnA\, (ρnB)r) <_ε. If \t\ > T
= \ρ\N then |ρ|/Ig|ί|<|ρ|π+1 for some n>N. Hence for each xε(tA}r\Dr we have
x = ta = (t/ρn)ρna with a e A and
Hence ρna e (ρ"A)nDr. From
we get
d(ρna,(ρnB)r)<s.
Since |f|Ξg |ρ|", we have
hence
So
hence
d(x,(tB)r) = d(ta,(tB)r)<\ρ\ε.
According to the above lemma,
,(tB)r)ί
sup
xe(tA)r\Dr
d(x,(tB)r)<\ρ\ε.
This proves δ((tA)r,(tB)r)-+Q while ί->oo. A similar analysis will give also
δ((tB\,(tA)r)-*Q while ί->oo.
2. Let ^4 be asymptotically ρ-self-similar about 0 with the limit model B. Fixe r > 0,
with (ρB)r = Br. Then (ρ^^β, and
lim ^(ρMUρ^H Mm δ((ρnA)r,Br) = Q
w-^oo
«-»oo
and
lim δ((ρnB)r,(ρM)r)= lim δ(Br, (ρnA)r) = 0 .
w~* oo
A, 5 verify (6) and then (7) by part 1.
w~* c»
Π
Similarity Between the Mandelbrot Set and Julia Sets
593
2
Proposition 2.3. Suppose that U, V are two neighborhoods of 0 in R and F:U^V
is a ^diffeomorphism with F(0) = 0 and with the derivative T0F=T non-singular.
Then for any closed set A C U the two sets TA and F(A) are asymptotically similar
about 0, i.e. for some r>0 (in fact for every r>0)
lim
ίe€,ί-» oo
Dr(tF(A),tTA) = Q.
Proof. Fix r >0. For all ε>0, then exists δ>0 such that if < \\x\\ <δ then
\\F(x)-Tx\\
\\x\\
For each yetTAnD,, there is xeA such that y = tTx with \\x\\ £\t~l\ \\T~~ *\\r
= C/\t\, where C = || T~1 ||r is a constant.
So for \t\>N = C/δ and any y = tTxe(tTA)r\Dr, yφO we have 0< \\x\\ <δ and
\\F(x)-Tx\\^ — — |ί|
<KtTx,tF(A))£\t\
Then by Lemma 2.1
~ "
' "' ')) r )^
SU
P
A similar analysis will give δ((tF(A))r9(tTA)r)->0.
Π
If F(A) is asymptotically self-similar then TA is so too. If in addition T
commutes with the similarity scale, A should be also asymptotically self-similar.
More precisely,
Proposition 2.4. Assume that F is as in Proposition 2.3. Suppose that A is a closed
set such that F(A) is asymptotically ρ-self-similar about 0 with the limit model B. If T
is fa-linear, then A is asymptotically ρ-self-similar about x, and the limit model is
T~VB, i.e. there is r'>0 such that
lim dr,(ρn(τ_xA\T-^B) = Q.
n-» oo
Proof. By the above proposition, TA and F(A) are similar about 0. Choose r > 0
such that
lim dt(QnTA9ρnF(A)) = 0
n-*oo
and
lim dr(ρ"F(A),B) = Q.
n—> oo
Then
lim dr(ρ"TA,B) = 0.
n~+ oo
Since Tis C-linear, we have ρ"TA = Tρ"A. For r'>0 such that Dr.CT'ίDr,
lim dr.(ρ"A,T-1B) = 0. D
594
TAN Lei
This proposition will be our essential tool in the next section to prove the selfsimilarity of Julia sets. The next proposition gives another way to produce new
self-similar sets:
Proposition 2.5. Suppose that X is a (asymptotically) ς>-self-similar set about 0.
Then the set
d
Y={zE<E\z EX}
is also (asymptotically) ρ-self-similar about 0. In fact for any ρ' such that ρfd = ρ, the
set Y is (asymptotically) ρ'-self-similar about 0.
Proof. We will only deal with the self-similar case. Choose r>0 such that
(ρX)rd=Xrd. Remark at first
z e ynD r/ | β /| <=> zd E XnDrd/(ρ'd| <=> ρzd e XnDrd
<=> (ρ'z)d e Xr\Dfd <s> Q'Z E YnDr.
Hence (ρ'Y)nDr = 7nDr and then (ρ'Y)r = Γr G
3. Self-similarity of Julia Sets
In this section we discuss the self-similarity of the Julia set of rational maps. And we
apply the result to the case of Misiurewicz points.
Definition and known results (cf. [D] and [DH1]).
1. Let /: <D-*(C be a rational map. A point x e <C is periodic of period p if p is the
minimal number such that fp(x) = x. The value ρ = (fp)'(x) is called the eigenvalue
of the orbit of x, it does not depend on x, but only on the orbit of x. A periodic point
x is said to be
(a) repulsive if |ρ| > 1,
(b) attractive if 0<|ρ|<l,
(c) superattractive if ρ=0,
(d) rational indifferent if ρ = e2πiθ, with θ rational,
(e) irrational indifferent if ρ = e2πiθ, with θ irrational.
2. A point x e C is eventually periodic if there are integers / ^ 0 and p ^ 1 such that
p l
l
f (f (x))=f (x)We say that x is eventually repulsive (attractive,..., etc.) periodic if
f\x) is repulsive (attractive,..., etc.).
3. A closed set A is said to be completely invariant under / if f(A) = f ~ί(A) = A. If
A is completely invariant under / it is completely invariant under fk as well.
4. The Julia set Jf of/ is the closure of the set of repulsive periodic points of/ We
know that Jf is completely invariant by/
5. For P a monic complex polynomial, the filled-in Julia set KP of P is
Kp = {z E CI P"(z), n e N, is bounded} .
We have JP = dKP and KP is completely invariant under P.
Similarity Between the Mandelbrot Set and Julia Sets
595
Remark. There are several equivalent definitions of the Julia set. The reason
that we choose this one is that repulsive periodic points play the central role in
the study of asymptotic similarity.
Concerning Misiurewicz points, we recall
Definition.
1. The family of quadratic polynomials can be parametrized by (up to affine
conjugacy)
Kc denotes the filled-in Julia set of fc and Jc the Julia set.
2. The Mandelbrot set is
M = {ce(C\QeKc}.
3. A point ceM is a Misiurewicz point if 0 is eventually periodic for fc but not
periodic.
A classical result of Douady and Hubbard shows:
Proportion 3.1 [DH1]. For c a Misiurewicz point,
1. 0 and then c is eventually repulsive periodic;
2. KC = JC, i.e. Kc has no interior.
Here comes the main theorem of this section:
Theorem 3.2. Let f be a rational map and A be a completely invariant closed set
under f. Assume that x is an eventually repulsive periodic point for f. Then A is
asymptotically self-similar about x, with the scale ρ equal to the eigenvalue of the
eventual periodic orbit of x. There is a conformal mapping φ defined in a neighborhood U of x such that ... . φ(AnU) is the limit model of A at x. Moreover, if x is
φ(x)
periodic, we can choose φ such that φ'(x) = \; if x is eventually periodic with
fp(f\x))=fl(x)
and if in addition, (fl)'(χ)ή=Q, then the limit models of A at x
l
and at f (x) are the same, up to a multiplication by (fl)'(x).
Hence not only A is asymptotically self-similar about x, but its limit model can
be realized locally (via a conformal mapping) instead of just asymptotically.
In the case where c is a Misiurewicz point, we can apply this theorem
immediately to A = Jc about x = c and claim that Jc is asymptotically self-similar
about c, since the point c is eventually repulsive under fc. Corollary 3.5 below will
give the precise formulas.
To prove Theorem 3.2, we need at first a classical lemma (see, for example,
[D]):
Lemma 3.3.
1. Suppose that U, V are neighborhoods of x in C and g: U-^V is a holomorphic
function with g(x) = x and \g'(x) \ φ 0, 1 . Set ρ = g'(x). Then g is locally conjugate to its
linear part z\-+ρz. Mote precisely, there is a conformal mapping φ defined in a
neighborhood of x with φ(x) = 0, φ'(x)= 1,
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TAN Lei
The explicit form of φ is given by
φ(z)= lim ρKg-'Πz)-*),
H-+00
1
where g' is the unique inverse mapping of g defined in a neighborhood of x. The
mapping φ is called the linearization mapping of g about x.
2. Suppose that U, V are neighborhoods of x, j; in <C respectively and g: U-+V is
holomorphic with g(x) = y and g'(x) = g"(x) = ...=g(d~ l\χ) = 0 and g(d\x) φ 0. Then
there are UXCU, VycV neighborhoods of x, y respectively, s > 0, such that for each
conformal mapping φy:Vy-*Dsd there is (with d choices) a conformal mapping
φx:Ux-+Ds satisfying
9y°g°9x\z)
= Zd.
Lemma 3.4. Assume that A is a completely invariant set of f. Then for any U" C (D, we
have
Proof. f(An U) C f(A)nf(U) = Anf(U). On the other hand, y e Ar\f(U) => y e A
and y = f(x) for some x e U. Since / ~ l(A) = A we have x e A and hence y ε f(UnA).
So we have also
Ar\f(U)Cf(AnU),
D
Proof (of Theorem 3.2).
a) Assume at first that x is periodic.
Let p be the period of x and ρ = (fp)'(x) be the eigenvalue of the orbit of x. By the
assumption, |ρ|>l. So fp arises to a homeomorphism in a neighborhood of x.
Remark that x is a fixed point of fp. According to Lemma 3.3 we can find a
neighborhood F of x, a neighborhood F0 of 0, a conformal mapping φ : F-» F0 such
that φ(x) = 0, φ'(x) = l and
Take r>0 such that D,cV0. Set U = φ~1(Dr/β) and B = φ(AnU). We have
ff(U) = φ- !(£),.) c V and U C f(U). Since A is completely invariant under f, from
Lemma 3.4, we have ΐp(AnU) = Anf(U). Hence
Then
φ(f(An C7)n U) = φ(ArΛ U) .
Since φ is bijective, we have
φ(f>(An U))nφ(U) = φ(A)nφ(U) ,
ρ(φ(AnU))nDr/β = φ(A)nDr/β ,
597
Similarity Between the Mandelbrot Set and Julia Sets
This proves (ρJ5)r/ρ = £r/ρ. Hence B is ρ-self-similar about 0. Apply Proposition 2.4
to φ:V-+V0, AnU->B; we conclude that A is asymptotically ρ-self-similar
about x, and the limit model is exactly
= B = φ(Ac\Ό).
φ'(x)
b) Assume now that x is eventually periodic.
Let /, p be minimal such that fp(fl(x))=f\x).
Set α = /f(x) and ρ = (/p)'(α). By
the assumption, |ρ| > 1. Applying the part a) to α, we can find a neighborhood l/α
of α, a conformal mapping φ such that φ(α) = 0, (p'(α) = l, the set X = φ(AnUΛ) is
ρ-self-similar and
n
lim dr>(ρ τ-ΛA,X) = Q
for some r'>0.
bl) Assume at first that the set (x,/(x), ...,/*(x)} does not contain any critical
point of /, in other words, (fl)'(x) φ_0.
Then fl maps a neighborhood_ί7x of x homeomorphically to a neighborhood
F α c Ua of α, and f \Ar\U^ = Ac\Va. Apply Proposition 2.4 to fl: UX-*VΛ'9 we get
1
lim dr
(8)
for some r >0. The theorem is then done by setting φ = φ°fl.
b2) Suppose now (/')'(*) = (/')"(*)= ... =(/!f-D(x)=0 and (/')(<ί>(x) φ 0.
Then by Lemma 3.3 there are Ux, φx, s, 0<s<r 1 / d such that
Set
ί
We have Y=φx(AnUx), since X = φo/'(y4nt7,c) = (φ,e(^n{7Λ.))' . According to
Proposition 2.5 the set Y is also ρ-self-similar about 0. The theorem is then given by
the next formula with φ = φx:
lim
, -- Y
φx(χ)
=0.
(9)
Proof of (9). Fix s>0. We prove first
n
n
For zeρ (τ-.xA)nDs, we have z = ρ (y — x) with
take n large enough such that yeAnUx. Then
and |j; — x|^s/|ρ| w . We can
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TAN Lei
while tt~κx).
A similar analysis will give
and then (9). Π
We are ready now to look at Misiurewicz points:
Corollary 3.5. Assume that cisa Misiurewicz point. Let /, p be the minimal integers
such that
Seί «=/»,<?=(//)'(«). Then
ί. |ρ|>l.
2. There is a conformal mapping φ defined in a neighborhood of α, with φ(α) = 0,
φ'(α) = l and
where φ is given by
lim
«->• oo
with f~p the local inverse of f/ in a neighborhood of α.
3. (/C')'(
4. There are r>Q,a neighborhood Vofa,a neighborhood U of c, such that fcl(U) = V
and
lim dr ρ"(τ_Λ), 77]
w oo
\
<P(Λn
\Jc)\C)
= lim dr ρ"(τ_cjc),
φoffanϋ)
=0.
(10)
Hence Jc is asymptotically self -similar about c with the scale ρ and the limit model
φ(JcnV).
Proof.
1. From Proposition 3.1.
2. The existence of φ is guaranteed by Lemma 3.3 if we replace g by fcp and x by α.
3. Since 0 is the unique critical point of fc and is not periodic, there is no critical
point of fc in
hence
4. It is a consequence of parts a) and bl) in the proof of Theorem 3.2, by setting
We give now some examples to illustrate the result of Theorem 3.2.
Similarity Between the Mandelbrot Set and Julia Sets
599
Examples
1. Figure 3b is the Julia set for c = 0.11031-0.67037ΐ. The polynomial fc
has two fixed points, both repulsive. We choose x = (l —1/1—4c)/2« —0.14205
— 0.52205/ to be one of them. The period p of x is then 1, ana the eigenvalue ρ of x is
2x& —0.2841 —1.0441 i. Figure 6 consists of a sequence of magnifications of
Jc at x, the magnification factor from one image to the next is ρ3 (since
|ρ|« 1.082 is too close to 1).
2. Figure 3c is the Julia set for c = — 1.25. It has a very different shape from Fig. 3b.
In fact one is a connected set and the other one is homeomorphic to a Cantor set.
But thefixedpoint x = (1 - |/l-4c)/2 which is (1 -j/6)/2 w - 0.72475 in this case is
again a repulsive fixed point for fc =/_ 1<25 . So the self-similar phenomenon occurs
again around this point. We have p = 1 and ρ « —1.4495. Figure 7 consists of three
magnifications of Jc at x, the magnification factor from one image to the next is
|ρ| = 1.4495. Since ρ is real negative, there is a difference of the rotation of angle
180° from one image to the next. Figure 7a and Fig. 7c then look the same.
3. Let us check the Misiurewicz point c = i now. For the polynomial f{: zt-+z2 + i,
the orbit of i is: i\-+i— 11-» — i\-*i — 1. The point i is eventually repulsive periodic,
with / = !, p = 2, α = /cz(c) = i-l,
ρ = (/C2)'(a) = 4 + 4ΐ = 4|/5e'"/4
and (/c')'(c) = φ'(0 = 2ί. Denote by X the limit model of Jt at / -1 and Z the limit
model of Jt at i. By our theorem, we have Z = (l/2ΐ)-X". Figure 8a is a magnification
of Fig. 3a in a neighborhood of i — 1. Figure 8b is a magnification of Fig. 8a with
5Λ&*3S
' 4f**H&**A
. •
Fig. 6. Three successive magnifications of Jc for c = 0.11031-0.67037Ϊ. Center: -0.14205
-0.52205Ϊ, width of the first picture: 0.01. Magnification factor: 1.26688, rotation: 44.3349°
Fig. 7. Three successive magnifications of Jc for c = -1.25. Center: -0.72475, width of the first
picture: 0.01. Magnification factor: 1.4495, rotation: 0
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TAN Lei
Fig.8a-d. Magnification of Jc for c = l a Center z — 1, widths: 0.001, rotation: 0. b Center i—1,
width: 0.001/4J/2, rotation: 45°. c Center i, with: 0.0005, rotation: 0. d Center i, width: 0.0005,
rotation: 90°
the factor 4j/2eπί/4. Figure 8c is a magnification of Fig. 3a in a neighborhood
of ϊ with the width of the window equal to the half of the width in Fig. 8a.
Figure 8d is Fig. 8c rotated by an angle of 90°. Compare Figs. 8a and 8d, one is
centered at i—1, the other one is centered at i. This computer experiment confirms by impression our result.
4. Key Proposition
This section will give a connection between the self-similarities of dynamic
spaces and parameter spaces, in a high-dimension form.
It is easy to generalize the notion of similarity to closed subsets of an euclidean
space of finite dimension. Let £ = Cfc and 2? be the set of compact subsets of E.
Denote by Dr the ball of E centered at 0 with radius r. Then the Hausdorff distance and the Hausdorff-Chabauty distance are defined in exactly the same
way. Moreover,
Definition.
1. A linear mapping L:£->£ is contracting if there are σ<l positive and C
positive such that for all neN, ||Ln|| ^Cσn. We say that L is expanding if
L is invertible and L~ 1 is contracting.
2. A subset B of E is ί^self-similar about 0 if L : E -> E is linear expanding and there
is r > 0 such that (LB)r = Br. We also say that B is self-similar about 0 with scale L.
3. A subset B of E is asymptotically L-self-similar about a point x e B if L : E-*E is
linear expanding and there is r>0, A^SF such that
lim (Lnτ.xB)r =
Similarity Between the Mandelbrot Set and Julia Sets
601
exists for the Hausdorff distance (where τα is the translation zi—>z + α). The set A is
automatically Z^self-similar about 0, it is called the limit model of B at x.
Here is our main result of this section:
Proposition 4.1. Suppose A is a neighborhood of λ0 in E, and XcAxE. For each
λeΔ, set
= {xeE\(λ,x)εX}.
Let u:A-+E be a continuous mapping, with u(λ0) = 0. Set
Assume:
M = Mu = { λ e A \ u ( λ ) ε X ( λ ) } .
(11)
Condition 1. X is closed in AxE.
Condition 2. (Existence of a dense set of continuous sections at (λQ, 0).) There is
A C X(λ0), dense in X(λQ), such that for each xeA, there exists UXCA, neighborhood
of λQ, and a continuous mapping hx:Ux-+E with hx(λ0) = x and hx(λ)eX(λ).
Condition 3. X(λ) is L(λ)-self-simίlar about 0 with a fixed radius r', and λ\-*L(λ) is
continuous, with
\\L(λ)-L(λ0)\\=0(\\λ\\)
and
||L(^)|HI#0ΓT<1,
(12)
we can assume then for λeA (by restricting A if necessary),
and
Condition 4. The derivative S = Tλou exists and is non-singular, and
2
\\u(λ)-Sλ\\=0(\\λ\\ ).
(13)
Then there is r>0 such that
a) Under the conditions 1, 2, we have dr(X(λ),X(λ0))-+Q when λ-*λ0.
b) Under the conditions 1, 2, 3, 4, the set Sτ _AoM is asymptotically L(λ^-self -similar
about 0, and the limit model is X(λ0). Precisely we have
c) // S and L(λQ) commute, then M is asymptotically L(λ0)-self -similar about λ0, and
the limit model is S~1X(λQ), i.e. there is s>0,
lim Mλorτ-^Λί^S-^μo)),.
(14)
Remark. Equation (12) is automatically verified if L(λ0) = ρl. Otherwise, let
{/!, /2, ..., lk} be the set of eigenvalues of L(λ0) and
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If|'fcl<IΊI 2 then there is a norm such that (12) holds. This requires that L(/10)
deforms the centered balls in a somehow homogeneous way. M. Shishikura
has a counter-example which shows that the proposition becomes false if the
condition (12) is not satisfied.
Condition 1 can be stated in an equivalent but more useful form:
Lemma 4.2. The following two conditions are equivalent:
1. X is a closed subset of A x E;
2. For each λ0εA, the set X(λQ) is closed in E and for all r>0, and lim δ(Xr(λ\
We leave the proof to the reader.
Proof (of Proposition 4.1). Without loss of generality, we may assume A 0 = 0.
a) By Condition 1 and Lemma 4.2, fix any r > 0 we have
Let us prove now
lim
λ-»0
by using the dense set of continuous sections of the Condition 2. Denote by Vr(ε) the
ε-neighborhood of dDr. Since Xr(G) is compact, and A is dense in X(ϋ) by
Condition 2, Vε > 0, there is a finite number of points xί9 x2, . . ., xm in A such that
Xr(0)GFr(ε/2)u
U
V=ι
where D(xi9 ε/2) is the ball centered at xt and with radius ε/2. Now for each i, 3ηt > 0
small enough such that when ||λ|| <ηt, we have
so
Hence Vε>0, for \\λ\\ <η = min{ηi}, VxeZ(0)nZ)r, we have
)£ \\x-Xi
for some i, hence δ(Xr(Q), Xr(λ}) < ε.
b) Set M(n) = L(0)n SM and let r>0 be smaller than r' required in Condition 3.
Our proof consists of two steps:
bl) δ(Mr(n),Xr(ϋ))-+Q when n-»oo.
We are going to prove that for each ε > 0, there is N > 0, such that for n > JV, and
for
L(Q)n
SλeM(n)nDr
we have
n
n
\\L(Q) Sλ-L(λ) u(λ)\\<ε/2
(15)
(16)
Similarity Between the Mandelbrot Set and Julia Sets
603
and as a consequence
(17)
Let y=L(0)n SλeM(n)nDr. Then λ=S~1L(OΓ*y So
IIAKHSΓ 1 !! ||L(OΓT r^O4 (C is a constant).
(18)
Proof of (15). We have
ί ||L(θr Sλ-HΰTu(λ)\\ + ||L(0)Xl)-L(Ar«(A)|| .
(19)
And by using Condition 3 and 4 and the inequality (18), we get
||L(θr SA-L(0)"u(A)|| ^ ||I(0)||" \\Sλ-u(λ)\\
(20)
V2 n = C5nμ" ,
(21)
where Ct are constants. Since lim μ" = lim nμ" = 0, from (19), (20), (21 ), we get (1 5).
«-> oo
n->oo
Proof of (16). From (15) we have
Choose ε sufficiently small such that r 4- ε < r'. Suppose A e M. We have u(λ) e
hence
L(λ)nu(λ)eX(λ).
If L(A)πM(/l)eFr(ε/2) [recall that 7r(ε/2) is the ε/2-neighborhood of δDJ, then
(16) is true. Otherwise
According to part a) <5(^(A), X^(0))^0 when λ->0. So to prove (16) we need to
show λ-»0 when n->oo. But this is guaranteed by (18).
From (15), (16), we get (17). In other words: Vε>0, 3ΛΓ>0, such that Vn>ΛΓ,
V;yeMr(tt), we have d(y,Xr(0))<s, hence δ(Mr(n\ Xr(Q))-+Q when n-^oo.
b2) (5(Zr(0),Mr(n))->0 when n-^oo.
b2.1) Let us prove at first that for each xeAnDr we have φc,Mr(n))-»0. By
Condition2, there is a continuous mapping hx:A-*E with ^(0) = * and
hx(λ) e X(λ\ Lemma 4.3 below will show that fixing any neighborhood Δ' C Δ of 0,
there is N' > 0, such that for each n > N', the next equation has at least one solution
inzΓ:
hx(λ) = 0,
(22)
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i.e. there is λne A', such that
(23)
Now we have automatically A π eM, and hence L(0)"Sλn e M(n). Therefore
Sλ\\
ϊ \\x-hx(λn)\\ + \\L(λnru(λn)-L(θγ Sλn\\ =/1(n) + /2(n).
(24)
Let us prove at first
I2(n)=\\L(λnru(λn)-L(0)"
when
Sλn\\^0
»-*oo.
Since L(Aπ)"u(An) = ftx(/ln), and /jχ is continuous, we can suppose
IIMAJII^ sup ||
for some r^O. Then
||Wμπ)|| = ||L(4)-"/zx(4)|| grJILίλJ-^r^r^.
(25)
By Condition 4 of the proposition, we have also
Take α< \\S\\C~~1, then for ||A|| ^α, we have
Hence
μ||^C 0 ||wμ)||.
(26)
Combining with (25), we get
(27)
(C,C0,C are constants). Hence for 24'cD(0,α), the solutions /lπ of (23) also verify
(18). By using the results of (19), (20), (21) we get that for ε>0 and n>max{]V, N'}9
the formula (15) holds for iπ, i.e.. /2(n)<ε/2.
From the continuity of hx and (27), we may also assume that under the same
condition for n,
Hence /1(n) + /2(n)-^0 when n->oo. From (24) we get finallly
d(x, Mr(ή)) -> 0
when
n -> oo .
b2.2) For the general case, let us do as in a). For each ε>0, choose {xj a finite
subset of AnDr such that
and choose n large enough such that Vx e Xr(ϋ), we have
Similarity Between the Mandelbrot Set and Julia Sets
605
for some i. Hence
when n-κx).
c) This is a direct result from part b), by taking 0<s<r/||S||. D
The lemma below we needed in the proof of the theorem is in fact a topological result, which says that under small perturbations a continuous mapping
does not lose zeros.
Lemma 4.3. Let A'cEbea neighborhood of 0, and u'.Δ'^Ebe continuous, u(0) = 0,
T0u = S exists and is non-singular. Then there is η>0 such for any w:A'-+E
continuous with ||w(x)|| <η, the mapping u + w has at least one zero in A'.
Proof. There is ε > 0, such that Sε = {x E E \ || x \\ = ε} is contained in A ', the set u(Sε)
does not contain 0 and the induced mapping
is an isomorphism (where k is the real dimension of E and Hj is the /h homology
group). Let
η = mϊ {\\u\\ \ueSε}.
Then for w : A ->E continuous and || vφc) || < η, we have u + w : Sε-+E — {0}, and the
induced mapping (u + w)^ is also an isomorphism, since u -f ίw gives a homotopy
between u and u + w. We claim then u + w has zeros in Dε = {x e E \ || x \\ ^ ε}. If not,
(u + w)* :Hk(Dε)^Hk(E — {0})=Z would be trivial homomorphism, since u + w is
continuous on Dε, and Hk(Dε) would be trivial. This is a contradiction. Π
To induce (22) from this lemma we choose N such that if n > N then
hx(λ)
Remark. If u and w are C-analytic, then under the same condition u + w has a
unique zero in Dε for small ε.
5. Similarity Between M and Jc for c a Misiurewicz Point
In this section we are going to apply the result of Sect. 4 to show the similarity
between the Mandelbrot set M and the Julia set Jc for c a Misiurewicz point.
Recall that under fc:z\-+z2 + c the two sets Kc, Jc are completely invariant.
Hence QeKc if and only if c = fc(0)eKc. We can then express the form of the
Mandelbrot set by
M = {ce€\ceKc}.
(28)
Recall that c 6 (C is a Misiurewicz point if 0 is eventually periodic for fc but not
periodic, and in this case KC = JC. According to Corollary 3.5, Jc is asymptotically
self-similar about c with a scale ρ and with a certain limit model Z. We will prove
that M is also asymptotically ρ-self-similar about c but with the limit model λZ9 for
a certain complex number λ. In other words,
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1. M is asymptotically self-similar about c;
2. M and Jc are asymptotically similar about c, up to a multiplication by a
complex number.
Or to be more explicit:
Theorem 5.1. Let c0 be a Misiurewicz point. Then there are ρ0 with |ρ 0 |>l, r, r',
s>0, a closed set ZcC with ρ0Z = Z, and Λe<C-{0} such that
lim (ρ5τ.CoJJΓ = Z r ,
(29)
lim (ρ»0τ_coM)r, = μZ)r,,
(30)
n-*oo
W-+QO
and
lim
ίeC,ί-*oo
ds(ίr_CoM,α τ_ C o J C o ) = 0.
(31)
The precise form of ρ0, λ, Z wz// be git en in (32), (37), and (36).
Proof. We will apply Proposition 4.1. Since Kc and Jc are not self-similar about c,
but only asymptotically self-similar, we have to adapt the situation at first.
Let / be the minimal number such that fcl0(c0) is periodic and let p be the period
of/c'0(c0). Seta 0 = /i(c0))and
(32)
Qo=(m*0).
We have |ρ0| > 1 and (/Co)'(c0)=t=0 and JCQ = KCQ. According to Theorem 3.2 the
invariant set KCQ is asymptotically ρ(c0)-self-similar about α0, with a certain limit
model X(cQ), and about c0 with the limit model Z =
t
—- X(c0).
C
CQ) ( θ)
Since α0 is a repulsive periodic point of fco, it is "stable" with respect to the
parameter. In fact, applying twice the implicit function theorem, we can find a
neighborhood W of c0 in (C and two analytic functions
v:W-»<C,
cι->v(c),
such that
:))ΦO,
and ρ: CH->ρ(c) = (//)'(α(c)) is analytic in W with |ρ(c)| > 1 for all c e W. According to
Theorem 3.2 the filled-in Julia set Kc is asymptotically ρ(c)-self-similar about α(c)
with a certain limit model X(c), and about v(c) with the limit model
t
X(c).
We need now an extension "with parameter" of Lemma 3.3:
Lemma 5.2 (high dimension case of Lemma 3.3). Suppose that U is a neighborhood
of (Λ),α0) in C / l + 1 =C / I xC. Let G:U-+<Cn +ί be a holomorphic function with
G(A,z) = (XgΛ(z)) and G(A0,α0) = (λ0,α0). Assume |gΆ 0 (α 0 )|>l. Then the sequence
of mappings
Φn: (λ, z) i—> (A, (gfλ(θί(λ)))"((gΐ1 )"(z) ~"α(^))) j
n = 1,2,...
Similarity Between the Mandelbrot Set and Julia Sets
607
converges to a holomorphic mapping Φ:UlL-+(£n+l, where Ui C U is a neighborhood
of (A0,α0), a(λ) is the implicit holomorphic solution of gλ(z) = z satisfying a(/l0) = a0
and g^1 is the local inverse of gλ. The mapping Φ has the following form
with φA(a(/ί)) = 0, φ'λ(a(λ)) = \ and
According to this lemma the linearization mapping φc of fc about α(c)
varies analytically with respect to c. In other words, the mapping
φ : (c, Z)H-»(C, φc(z)) ,
(c0, α0)h-»(c0, 0)
is analytic in a neighborhood of (c0,α0). Since (c,x)t-+(c,fcl(xj) is also analytic, the
mapping
Φ : (c, Z)M>(C, φc o fc\z)) ,
(c0, CO)H>(CO? 0)
is analytic in a neighborhood U of (c0, c0).
Choose r>0 and restrict W if necessary such that W is closed and
Φ(c,c)eDrxWcΦ(U).
Set consequently
K = {(c,z)\zεKc},
X(c) = φeof*(KenΩJ,
and
X = Φ(KnΩ) = {(c, x) I c e W, x e X(c)} .
Then Ω is closed in C x C, Ωc is closed for each ceW, and
for each
cePF.
(33)
Moreover, we possess an important property of K proved by Douady and
Hubbard:
Proposition 5.2 ([DH1], expose no. VIII). The set K is closed in (C x C.
Now we are ready to apply Proposition 4.1 to Xc Wx C. Set £ = <C, A = W,
and
Mu = {cEW\u(c)EX(c)}.
Then w(c0) = 0 and
Mu = [c 6 W I φc o /Λc) e φc °fcl(KcnΩc)} .
But from (33) we have ceΩc as soon as ce VK so
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Hence Mu is exactly the Mandelbrot set M in a neighborhood of c0. Let us checck
the conditions of Proposition 4.1 now:
For Condition 1, since K is closed (Proposition 5.3), Ω is closed, Φ is analytic so
the set X = Φ(KnΩ) is closed in (C x C as well.
For Condition 2, we have at first JCQ = KCQ, that repulsive periodic points are
dense in Jco, and that each repulsive periodic point x is "stable" with respect to c. In
fact, by the implicit function theorem, there is a neighborhood Ux of c0, an analytic
mapping hx:Ux->1C with hx(c0) = x and hx(c) a repulsive periodic point of fc.
Hence hx(c)eJccKc. Hence there is a dense set of continuous sections in K at
(c0, c0), so does the set X at (c0, 0). This gives Condition 2 for X.
For Condition 3, set L(c) = ρ(c)I. Then X(c) is L(c)-self-similar at 0, and
\\L(c)-L(c0)\\=\ρ(c)-ρ(c0)\ = 0(\c-c0\),
Condition 3 is then well verified.
For Condition 4, since u is analytic, Eq. (13) is automatically verified. To check
U'(CQ) φ 0 is the main difficulty here. We will do that in Lemma 5.4 below, by using a
non-trivial result of Douady and Hubbard. Moreover, we will get the following
explicit form of u'(c0):
"'(c0) = ~ (φc(f!(c))) |c=co = A (/c<(c)) |c=co _ *. (α(c))Uo .
(34)
Hence all the conditions of Proposition 4.1 are verified. From (14) we can
claim then that Mr\W=Mu is ρ(c0)-self-similar about c0 and the limit model
is —;—- X(c0), i.e. there is r > 0 such that
lim (ρ(co)"τ-coΛf)r=
~ *(c0)
(35)
Recall that X(c0) = φco <> fc[(KCo) is the limit model of Kco = JCQ at α(c0) = fel0(c0)9
and
—- X(c0) is the limit model of KCQ at c0. In other words, set
f
Uc0) (co)
z=
x(Co)=
φΛo)
φ
«°&κ Jss
(36)
we have:
lim (ρSτ_CoJCo)r = Z r .
That is (29). In (35) substitute w'(c0) by its explicit form (34) and X(c0) by
we get finally (probably for a r'< r):
lim
(fclJ(c0)Z,
(ρ(cn)nτ_ΓM)r>
This is Eq. (30) of the theorem, with
/ f l \ff
\
.
(37)
609
Similarity Between the Mandelbrot Set and Julia Sets
2.2.
Equation (31) is a direct consequence of (30) and (29) according to Proposition
Π
l
Lemma 5.4. For u: PF->C, c\-*φc(fc (c)), we have
U (Co)=
'
(38)
c
Tc tfΛ »l«=«o- (α(c))|c=co
and
«'(C0)ΦO.
Proof. Set β: C-+C, c^fcl(c)=fcl+l(Q) and w: C^C, c^fcl+1+p(0)-/cz+^O). We
have jβ(c0) = α0 and w(c0) = 0. We will see
and w'(c0)φO.
Set F(c, z) = φc(z), where φc is the linearization mapping of /c about α(c). Then
f
Since
dF_
~dz
(co, Λ
(CD, αo
dF
(co.αo)
c-c0
c-cn
'a'(c0)=-a'(cQ)
/
/
/
[since F(c0,α0) = F(c,α(c))-0], we have then M (c0) = j8 (c0)-α (c0);
,.
w(c)-w(c0) _ ,.
C
^c)-α(c)
The fact w'(c0) Φ 0 is not evident and is proved by Douady and Hubbard. They
have in fact several proofs about that. One is stated in the last corollary of [DH1],
another one is an arithmetic method and is inserted in [DH3] (Lemma 1 of
Chapter V). Π
Theorem 5.5 (Review). Let us summarize now the main steps to prove the similarity
between M and Jc. That is, the main steps of the adaptation of the Misiuremcz case
to the assumptions of Proposition 4.1.
Assume that c0 is eventually repulsive periodic under the iteration of fco (i.e. c0 is
a Misiurewicz point). Let l,p be the minimal integers such that fcp0(fcl0(c0)) = fclQ(c0).
Set α0 = fcl0(Co) and ρ0 = (f£)(&$). Then for c varying in a neighborhood W of c0 in (C:
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1 . There is a unique repulsive periodic point α(c) of period p for fc such that c i—> α(c)
is analytic and α(c0) = α 0 ; let ρ(c) == (/cp)'(α(c)), the mapping cι—>ρ(c) is analytic with
Q(CQ) = QQ (implicit function theorem);
2. Kc is asymptotically ρ(c)-selj '-similar about a(c) with a certain limit model X(c\
provided that it is completely invariant under fc (Theorem 3.2);
3. There exists a conformal mapping (the linearization mapping) φc defined in a
neighborhood of α(c) in C with φc(α(c)) = 0, φfc(oc(c))=l, φc(K^ = X(c\ and
(c,z)ι—>(c, φc(z)) is analytic in a neighborhood of (c0,α0) in C x C (Lemma 3.3,
Theorem 3. 2 and Lemma 5.2);
4. The set K = {(c,z)\zeKc} is closed in Wxdu (Proposition 5.3). This gives
Condition! of Proposition 4. i;
5. KCO = JCO (Proposition 3.1). This gives Condition 2 of Proposition 4. i, provided
that repulsive periodic points are dense in JCQ;
6. Condition 3 of Proposition 4.1 is a consequence of 2 and 3;
1. For u:c^φc(fc\c)\ we have MnW={ceV\u(c)eX(c)} = Mu;
8. u is analytic and w'(c0)Φθ (Lemma 5.4). This gives Condition 4 of
Proposition 4.1;
9. M is asymptotically ρQ-self-sίmίlar
about c0 with the limit model ——- X(c0)
U(CQ)
(Proposition 4.1);
10. (/Cl0)'(c0)φ0 and KCQ about c0 is asymptotically ρ0-self -similar, with the limit
model -±— X(c0) (Theorem 3.2);
(Jco)
C
( θ)
11. Finally M about c0 and KCQ about c0 are asymptotically similar, up to a
multiplication by
,
u(c0)
(consequence of 9 and 10 and Proposition 2.2).
To simplify the notation, denote by X(cQ\ Y(c0) and Z(c0) the limit model of KCo
at α0, of M at c0 and of KCΌ at c0 respectively. We have Y(c0)= -τr~\ x(co)
an(
i
Examples
1. Two spirals. We see a lot of spirals in M. The point c& —0.77568377
+ 0.13646737ί is a Misiurewicz point which is chosen so that M has a two spiral
shape near c. Figure 9a is a magnification of Jc around c and Fig. 9b is a
magnification of M around c.
2. Triple point. The point c= -0.1011 +0.95629Ϊ is the 3-fold bifurcation point
near the top of the Mandelbrot set. It is also a typical Misiurewicz point. In this
case, / = 3, p = l,
α = (1/2) (1 - j/l-4e) « - 0.3276 + 0.57776i ,
ρ = 2α« -0.6552 + 1.155524,
u\c) « - 4.751 5 + 2.06497/ ,
3
a = (/C )'(c) « - 6.4464 + 0.1 808i .
;
In other words: |ρ|wl.328, arg(ρ)« 119.6°, |ιι (c)|«5.1808, arg(ιι'(c)) = 156.51°,
\a\ ^ 6.4489, and arg(α) = 178.39°. Figures lOa are two successive magnifications of
Similarity Between the Mandelbrot Set and Julia Sets
611
Fig.9a,b. Magnifications of Jc and of M for c= -0.77568377 + 0.13646737Ϊ. a Jc, center: c,
width: 0.00018. b M, center: c, width: 0.00024
Fig. lOa-d. Magnifications of Jc and M around c = -0.1011 4- 0.9563i. a Jc, center: α, width: 0.01,
rotation: 0. b Jc, center: α, width: 0.00753, rotation: 119.6°. c Jc, center: c, width: 0.00155, rotation:
178.39°. d M, center: c, width 0.00193, rotation: 156.51°
Jc centered at α with the magnification factors differed by ρ. Figure lOc =Z(c)
= X(c)/u. Figure lOd = Y(c) = X(c)/u'(c). With these carefully chosen factors, we
can not distinguish M and Jc any more.
2
3. Let us take the example of c = i again. Recall that for f{ : zt-»z + i, we have / = 1 ,
πι /4
l
p = 2, α = i-l, ρ = 4|/2έ? ' , and (fc )'(c) = (fi)'(ί) = 2i. For the value of A, we have
The function α(c) is the solution near i — 1 of the implicit equation
for c close to i. Hence
2
(i- 1) + 0 2(i- 1)-
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Fig. 11 a, b. Magnifications of J{ and M around i. a Jί5 center: z, width: 0.0001, rotation: 0. b M,
center: i, width: 0.0001 x j/5/2; rotation: -26.565°
Then
2ί
2/-1
" 4i + 3
' 2
In other words, |>l| = J/5/2«1.118 and arg(/l)^26°565. Figure 11 shows the
similarity between M and J{.
Figure 12 generated by Peitgen, Jurgens, and Sauper ([PJS]) shows the selfsimilarity of M about i in another way. By our result, the set M should be almost
unchanged if we make two powerful blowups centered at i with the factors differed
by 4]/2 and by a rotation π/4. If we just blow it up successively by 4j/2 without
rotating, each one would differ by a π/4 rotation from the last one. Hence after
eight magnifications, we would see almost the same set. Compare for example
Fig. 12-4 and Fig. 12-12. Do you see any difference?
It seems that M does not spiral at i. But that is our human eyes' mistake. The
rotation angle π/4 explains well that there is a spiral there. The reason that we can
not "see" it is because the absolute value of ρ which is 4|/2 is too large with respect
to its angle π/4, so that the spiral is "absorbed" by the center. For example, take a
point x on the positive real axis. Then x/ρ8 is again a positive real number but its
value is decreased by a factor 1/224! It is so close to 0 that human eyes cannot
distinguish them any more. As an experiment, we take the spiral
A(s) = {e(lo*s+πi'4}x I - oo < x < + oo }
which is blowing-up-invariant by the factor ρ = seπιf4. The two pictures in Fig. 5
(s= 1.2 and s =4|/2) show how the spiral "disappears" as s increases.
6. Other Results
To end this paper we give a rough description of some other similarity problems.
We will see that for our definition of similarity the Misiurewicz points are the only
interesting case.
6.1. Small Copies of M Converging to a Misiurewicz Point. By looking at the
Mandelbrot set one realizes easily that it contains a lot of copies of itself. Eckmann
613
Similarity between the Mandelbrot Set and Julia Sets
r3
r 11
12
Fig. 12. 12 successive magnifications of M as indicated by the inserted frames. Center: i. Width of
the first picture: 0.5656854^0.1 x 4J/2. Magnification factor: 4]/2
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and Epstein ([EE]) and Douady and Hubbard ([DH3]) have studied the existence
and the scaling property of small copies of M near a Misiurewicz point. More
precisely, for each Misiurewicz point c e M , there is a sequence of copies Mn C M of
M converging to c geometrically with the ratio 1/ρ (where ρ is equal to our
similarity scale for M about c\ and the diameter of Mn decreases to 0 also
geometrically, but with the ratio l/|ρ|2. Hence diam(Mπ) decreases faster than the
convergence of Mn to c. This is compatible with our result that the limit model of
M at c has no interior point.
6.2. Some Trivial Cases. Using our definition of similarity one can check about
other points too. But sometimes the situation is more or less trivial so that it might
be better to analyze other notions of similarities.
Here is a list of the trivial cases:
1. Attractive periodic points in Kf and hyperbolic components of M.
Suppose that / is a monic polynomial and x is an eventually (super)attractive
periodic point. Then xeKf and Kf is self-similar about x. In fact Kf is trivially selfsimilar about any point y of Kf, since for r small enough we have (τ _yKf)s = Ds for
every s < r.
Similarly, the Mandelbrot set M is trivially self-similar about any point ceM.
Specially, for c in a hyperbolic component of M [i.e. fc has an (super)attractive
period cycle], M and Jc are trivially similar about c.
2. Rational indifferent periodic points in Jf and parabolic points in M.
First of all we observe that any two curves tangent at a point x are
asymptotically similar about x. Furthermore any closed set in C limited by two
tangent curves is asymptotically self-similar about that point, with the common
tangent line as the limit model.
Suppose that x is a rational indifferent periodic point for fc (for example, x = α
in Fig. 3d). Then xe Jc and there are finite petals in Kc\j{x] located one-by-one
tangently around x ([B, DH2]). Hence the Julia set is limited by tangent curves and
is asymptotically self-similar about x.
A point c e (C is called parabolic if fc has a rational indifferent cycle. Then
cedM and c is in the boundary of either one or two hyperbolic components
([DH2]). If c is on the boundary of two hyperbolic components, the two
components are tangent at c with tangent order two ([DH2, Tl]), consequently
dM and M are asymptotically self-similar about c. If c is on the boundary of only
one hyperbolic component W of M, dW has a cusp at c ([DH2]) so dM in a
neighborhood of c is again limited by tangent curves and then is asymptotically
self-similar about c.
6.3. Feigenbaum Points. The Feigenbaum points form a different set of boundary
points of M. The classical Feigenbaum point is the limit point of the period
doubling sequence on the real axis starting at c=— 1, which is first studied
experimentally by Feigenbaum and GroBmann and Thomae. According to a
theory of Douady and Hubbard ([DH3]) M contains infinitely many small copies
of itself. Moreover, for each point ceM such that fc has a superattractive orbit,
there is a homeomorphism
with ιp(0) = c and δ(clΛί) C dM. The point clx is called the "tuning" of x by c. We
can then consider the sequence c, c_Lc, c_Lc_Lc, . . . and see if it converges, i.e. if the
infinite tuning exists. We call the eventual limit point a Feigenbaum point. For
example the above classical Feigenbaum point corresponds to the infinite tuning
Similarity Between the Mandelbrot Set and Julia Sets
615
Fig. 13. M and Jc around c, where c= - 0.745429 + 0.113008/. See also [PS]
of c = — 1. Feigenbaum has conjectured that the limit point for c = — 1 exists and
the sequence c,c_Lc, cJ_c_Lc,... converges geometrically. A more general conjecture is:
Conjecture 1. For every point c such that fc has a superattractive orbit, the
sequence
c, c_Lc, cJ_c_Lc,...
converges geometrically.
Some special cases of the conjecture including the case c = — 1 have been well
studied and confirmed. Here is a list of references: [CE, CEL, EEW, EW, GSK, and
L].
Denote by c100 the limit point and δ the inverse of the convergent-ratio. We
have \δ\>ί. Milnor ([M]) has made some computer experiments on the local
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structure of M about clco and has suggested that M should be asymptotically
measurely self-similar at clo° with the scale δ. In other words,
Conjecture 2. There is a measurable set X such that (δn(τ^c^M))r converges
for the Lebesgue measure to X. The set X is not closed, is everywhere dense but
may have small measure.
The fact that X should not be closed excludes any application of the
Hausdorff-Chabauty distance.
6.4. Peitgerfs Observation. H.-O. Peitgen ([PS]) has observed experimentally
another similarity phenomenon between M and Kc (Fig. 13): lor some value of c a
magnification with a carefully chosen power (neither too small nor too large) of M
and Jc about c gives very similar images. This phenomenon is different from what
we did in this paper because the means of similarity is rather local but not
asymptotic.
References
[B]
[CE]
[CEL]
[D]
[DH1]
[DH2]
[DH3]
[EE]
[EEW]
[EW]
[Fl]
[F2]
[GSK]
[GT]
[L]
[M]
[PR]
[PS]
Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. Am. Soc. 11,
85-141 (1984)
Campanino, M., Epstein, H.: On the existence of Feigenbaum's fixed point. Commun.
Math. Phys. 79, 226-302 (1981)
Collet, P., Eckmann, J.-P., Lanford III, O.E.: Universal properties of maps on an
interval. Commun. Math. Phys. 76, 211-254 (1980)
Douady, A.: Systeme dynamiques holomorphes, expose no. 599, seminaire Bourbaki
1982/83. Asterisque 105-106, 39-63 (1983)
Douady, A., Hubbard, J.H.: Etude dynamique des polynόmes complexes. Part I.
Publication mathematique dΌrsay, 84-02, 1984
Douady, A., Hubbard, J.H.: Etude dynamique des polynόmes complexes. Part II.
Publication mathematique d'Orsay, Orsay, 85-02, 1985
Douady, A., Hubbard, J.H.: On the dynamics of polynomial-like mappings. Ann. Scient.
EC. Norm. Sup, 4e serie, t. 18, 287-343 (1985)
Eckmann, J.-P, Epstein, H.: Scaling of Mandelbrot sets generated by critical point
preperiodicity. Commun. Math. Phys. 101, 283-289 (1985)
Eckmann, J.-P, Epstein, H, Wittwer, P.: Fixed points of Feigenbaum's type for the
equation fp(λx) = λf(x). Commun. Math. Phys. 93, 495-516 (1984)
Eckmann, J.-P, Wittwer, P.: A complete proof of the Feigenbaum conjectures. J. Stat.
Phys. 46, Nos.3/4, 455-475 (1987)
Feigenbaum, M.: The universal metric properties of nonlinear transformations. J. Stat.
Phys. 21, 669-706 (1979)
Feigenbaum, M.: Universal behavior in nonlinear systems. Los Alamos Sci. 1, 4-27
(1980)
GoΓberg, A.I, Sinai, Ya.G, Khanin, K.M.: Universal pronerties fe , laences of
bifurcations of period three. Russ. Math. Surv. 38, 187-188 (1983)
GroBmann, S, Thomae, S.: Invariant distributions and stationary correlation
functions of one-dimensional discrete processes. Z. Naturforsch. 32a, 1353-1363 (1977)
Lanford III, O.E.: A computer-assisted proof of the Feigenbaum conjectures. Bull, (new
series) Am. Soc, Vol. 6, Number 3, 427-434 (1982)
Milnor, J.: Self-similarity and hairiness in the Mandelbrot set. In Computers in
Geometry and Topology. Martin, C. (ed.). Tangora. New York, Basel: Dekker,
pp. 211-257 1989
Peitgen, H.-O, Richter, P.H.: The beauty of fractals. Berlin, Heidelberg, New York:
Springer 1986
Peitgen, H.-O, Sauper, D. (eds.): The science of fractal images. Berlin, Heidelberg, New
York: Springer 1988
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[PJS]
[Tl]
[T2]
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Peitgen, H.-O., Jϋrgens, H., Sauper, D.: The Mandelbrot set: a paradigm for
experimental mathematics, ECM/87-Educational Computing in Mathematics,
Banchoff,T.F. et al. (eds.), Elsevier Science Publishers BV. (North-Holland), 1988
Tan Lei: Ordre du contact des composantes hyperboliques de M, in [DH2], expose
no. XV
Tan Lei: Ressemblence entre Γensemble de Mandelbrot et Γensemble de Julia au
voisinage d'un point de Misiurewicz, in [DH2], expose no. XXIII
Communicated by J.-P. Eckmann